Quadripartite bond length rule applied to two prototypical aromatic and antiaromatic molecules

Context In 2000, a remarkably simple relationship was introduced, which connected the calculated geometries of isomolecular states of three different multiplicities. These encompass a ground single state, the first excited triplet state, as well as related radical anion and radical cation. The rule allows the prediction of the geometry of one of the species if the three remaining ones are known. Here, we verify the applicability of this bond length rule for two small planar cyclic organic molecules, i.e., benzene and cyclobutadiene, which stand as prototypical examples of, respectively, aromatic and antiaromatic systems. We see that the rule works fairly well to benzene, and it works independently for quinoid as well as for anti-quinoid minima, despite the fact that radical anion species poses challenges for correct theoretical description. Methods To obtain chosen electronic state equilibrium geometries, three types of computational approaches were utilized: fast and efficient density functional theory DFT, the coupled cluster method CC2, the complete active space self-consistent field (CASSCF) approach, and the most accurate but also resource-consuming perturbation theory with multireference wavefunction (CASPT2) with a default value and without IPEA-shift. Dunning and co-workers correlation-consistent basis sets (aug-)cc-pVXZ (X = D, T, Q) were employed. Gaussian 16 revision A.03, Turbomole 7.1, and Molcas 8.0 computational software were used. Supplementary information The online version contains supplementary material available at 10.1007/s00894-023-05498-4.

Selected geometrical parameters of anionic form of BZ(Q) S3 Table S2. Selected geometrical parameters of anionic form of BZ(AQ) S4 Table S3. Selected geometrical parameters of cationic form of BZ(Q) S5 Table S4. Selected geometrical parameters of cationic form of BZ(AQ) S6 Table S5. Selected geometrical parameters of ground state neutral form of BZ S7 Table S6. Selected geometrical parameters of first triplet excited state neutral form of BZ(Q) S8 Table S7.
Selected geometrical parameters of first triplet excited state neutral form of BZ(AQ) S9 Table S8. ΔGAH(R) values for selected bonds of BZ(Q) S10 Table S9. ΔGAH(R) values for selected bonds of BZ(AQ) S11 Table S10. Selected geometrical parameters of anionic form of CBDE S12 Table S11. Selected geometrical parameters of cationic form of CBDE S13 Table S12. Selected geometrical parameters of ground state neutral form of CBDE S14 Table S13. Selected geometrical parameters of first triplet excited state neutral form of CBDE S15 Table S14. ΔGAH(R) values for selected bonds of CBDE S16 Table S15. Selected geometrical parameters of neutral radical form of CP S17 Table S16. Selected geometrical parameters of bicationic radical form of CP S17 Table S17. Selected geometrical parameters of ground state cationic form of CP S18 Table S18. Selected geometrical parameters of first triplet excited state cationic form of CP S18 Table S19. Mulliken atomic spin densities for anionic form of BZ(Q) S19 Table S20. Mulliken atomic spin densities for anionic form of BZ(AQ) S20 Table S21. Mulliken atomic spin densities for cationic form of BZ(Q) S21 Table S22. Mulliken atomic spin densities for cationic form of BZ(AQ) S22 Table S23. Mulliken atomic spin densities for first triplet excited state neutral form of BZ(Q) S23 Table S24. Mulliken atomic spin densities for first triplet excited state neutral form of BZ(AQ) S24 Table S25. GAH-rule based atomic spin densities for ground state neutral BZ (from Q-forms) S25 Table S26. GAH-rule based atomic spin densities for ground state neutral BZ (from AQ-forms) S26 Table S27. Mulliken atomic spin densities for anionic form of CBDE S27 Table S28. Mulliken atomic spin densities for cationic form of CBDE S28 Table S29. Mulliken atomic spin densities for first triplet excited state neutral form of CBDE S29 Table S30. GAH-rule based Mulliken atomic spin densities for ground state neutral CBDE S30 Figure S1.
Maximum unsigned values of ΔGAH(R) values on the background of statistic ranges max(R)-min(R) S31 Table S1. Selected bond lengths [Å] and valence bond angles [ o ] between carbon atoms in quinoid-like anionic variant of benzene ring obtained from chosen approaches of quantum chemistry computational methods. C-C bond are labelled by designations introduced by Figure 1. If it is indicated, vibrational analysis was carried out for the equilibrium structure. All obtained structures are flat, possible imaginary frequencies relate to the swing of hydrogen atoms off the plane of symmetry, or are numerical artefacts of DFT methodology. S4 Table S2. Selected bond lengths [Å] and valence bond angles [ o ] between carbon atoms in anti-quinoid-like anionic variant of benzene ring obtained from chosen approaches of quantum chemistry computational methods. C-C bond are labelled by designations introduced by Figure 1. If it is indicated, vibrational analysis was carried out for the equilibrium structure. All obtained structures are flat, possible imaginary frequencies relate to the swing of hydrogen atoms off the plane of symmetry, or are numerical artefacts of DFT methodology.    between carbon atoms in antiquinoid-like cationic variant of benzene ring obtained from chosen approaches of quantum chemistry computational methods. C-C bond are labelled by designations introduced by Figure 1. If it is indicated, vibrational analysis was carried out for the equilibrium structure. All obtained structures are flat, possible imaginary frequencies relate to the swing of hydrogen atoms off the plane of symmetry, or are numerical artefacts of DFT methodology.    between carbon atoms in antiquinoid-like neutral variant of benzene ring in the first electronic triplet excided state obtained from chosen approaches of quantum chemistry computational methods. C-C bond are labelled by designations introduced by Figure 1. If it is indicated, vibrational analysis was carried out for the equilibrium structure. All obtained structures are flat, possible imaginary frequencies relate to the swing of hydrogen atoms off the plane of symmetry, or are numerical artefacts of DFT methodology. Imag. freq. S10 Table S8. Grochala, Albrecht, and Hoffmann Bond Length Rule formula values for selected carbon-carbon bonds [Å] and valence bond angles [ o ] in quinoid-like benzene variants. C-C bonds are labelled by designations introduced by Figure 1. The GAH rule is fulfilled more precisely when the value calculated according the formula is closer to zero. The charge of the molecule and its spin multiplicity can affect each bond length. Therefore unsigned relative percentage values are given in brackets. They were calculated by dividing the absolute value between the difference in length of a given bond in its longest and shortest form.  in anti-quinoid-like benzene variants. C-C bonds are labelled by designations introduced by Figure 1. The GAH rule is fulfilled more precisely when the value calculated according the formula is closer to zero. The charge of the molecule and its spin multiplicity can affect each bond length. Therefore, absolute values of relative percentage values are given in brackets. They were calculated by dividing the absolute value between the difference in length of a given bond in its longest and shortest form. TBDto be determined.  Table S10. Selected bond lengths [Å] between carbon atoms in anionic variant of cyclobutadiene (CBDE) ring obtained from chosen approaches of quantum chemistry computational methods. C-C bond are labelled by designations introduced by Figure 1.
If it is indicated, vibrational analysis was carried out for the equilibrium structure. All obtained structures are flat, possible imaginary frequencies relate to the swing of hydrogen atoms off the plane of symmetry, or are numerical artefacts of DFT methodology. All internal angles in carbon ring are right.  Table S11. Selected bonds lengths [Å] between carbon atoms in cationic variant of cyclobutadiene (CBDE) ring obtained from chosen approaches of quantum chemistry computational methods C-C bond are labelled by designations introduced by Figure 1.
If it is indicated, vibrational analysis was carried out for the equilibrium structure. All obtained structures are flat. All internal angles in carbon ring are right.    for carbon-carbon bonds in cyclobutadiene variants. C-C bonds are labelled by designations introduced by Figure 1. The GAH rule is fulfilled more precisely when the value calculated according the formula is closer to zero. The charge of the molecule and its spin multiplicity can affect each bond length. Therefore unsigned relative percentage values are given in brackets. They were calculated by dividing the absolute value between the difference in length of a given bond in its longest and shortest form.   Table S19. Mulliken atomic spin densities obtained with chosen computational approaches for quinoid isomer of anionic form of benzene molecule. In the right part of the table atomic spin densities of hydrogens are summed into heavy atoms they are connected to. Please note, that data for CASPT2 method are computed for equilibrium geometries obtained at this level of theory, however spin densities are computed basing on CASSCF wavefunction.  Table S20. Mulliken atomic spin densities obtained with chosen computational approaches for antiquinoid isomer of anionic form of benzene molecule. In the right part of the table atomic spin densities of hydrogens are summed into heavy atoms they are connected to. Please note, that data for CASPT2 method are computed for equilibrium geometries obtained at this level of theory, however spin densities are computed basing on CASSCF wavefunction. TBDto be determined.    Table S22. Mulliken atomic spin densities obtained with chosen computational approaches for antiquinoid isomer of cationic form of benzene molecule. In the right part of the table atomic spin densities of hydrogens are summed into heavy atoms they are connected to. Please note, that data for CASPT2 method are computed for equilibrium geometries obtained at this level of theory, however spin densities are computed basing on CASSCF wavefunction.  Table S23. Mulliken atomic spin densities obtained with chosen computational approaches for quinoid isomer of benzene molecule in the first electronic triplet excided state. In the right part of the table atomic spin densities of hydrogens are summed into heavy atoms they are connected to. Please note, that data for CASPT2 method are computed for equilibrium geometries obtained at this level of theory, however spin densities are computed basing on CASSCF wavefunction. Method  Basis set  H1  C2  C3  H4  C5  H6  C7  H8  C9  H10  C11  H12  H1+C2 C3+H4 C5+H6 C7+H8 C9+J10 Table S24. Mulliken atomic spin densities obtained with chosen computational approaches for antiquinoid isomer of benzene molecule in the first electronic triplet excided state. In the right part of the table atomic spin densities of hydrogens are summed into heavy atoms they are connected to. Please note, that data for CASPT2 method are computed for equilibrium geometries obtained at this level of theory, however spin densities are computed basing on CASSCF wavefunction. Method  Basis set  H1  C2  C3  H4  C5  H6  C7  H8  C9  H10  C11  H12  H1+C2 C3+H4 C5+H6 C7+H8 C9+J10 Table S25. Mulliken atomic spin densities for benzene molecule in the ground electronic state predicted as combination (R -+ R + -R 0 T1) of atomic spin densities computed for other electronic states structures in their quinoid variants. In the right part of the   Table S26. Mulliken atomic spin densities for benzene molecule in the ground electronic state predicted as combination (R -+ R + -R 0 T1) of atomic spin densities computed for other electronic states structures in their antiquinoid variants. In the right part of the table there are data for summed spin densities of heavy atoms and hydrogens connected to them. Please note, that data for CASPT2 method are computed for equilibrium geometries obtained at this level of theory, however spin densities are computed basing on CASSCF wavefunction. TBDto be determined.

BZ(AQ) R 0 T1
BZ(AQ) R -+ R + -R 0 T1 Method  Basis set  H1  C2  C3  H4  C5  H6  C7  H8  C9  H10  C11  H12  H1+C2  C3+H4  C5+H6  C7+H8  C9+J10 Table S27. Mulliken atomic spin densities obtained with chosen computational approaches for anionic form of cyclobutadiene. In the right part of the table atomic spin densities of hydrogens are summed into heavy atoms they are connected to. Please note, that data for CASPT2 method are computed for equilibrium geometries obtained at this level of theory, however spin densities are computed basing on CASSCF wavefunction.